Worst-case global optimization of black-box functions through Kriging and relaxation

Abstract

International audienceA new algorithm is proposed to deal with the worst-case optimization of black-box functions evaluated through costly computer simulations. The input variables of these computer experiments are assumed to be of two types. Control variables must be tuned while environmental variables have an undesirable effect, to which the design of the control variables should be robust. The algorithm to be proposed searches for a minimax solution, i.e., values of the control variables that minimize the maximum of the objective function with respect to the environmental variables. The problem is particularly difficult when the control and environmental variables live in continuous spaces. Combining a relaxation procedure with Kriging-based optimization makes it possible to deal with the continuity of the variables and the fact that no analytical expression of the objective function is available in most real-case problems. Numerical experiments are conducted to assess the accuracy and efficiency of the algorithm, both on analytical test functions with known results and on an engineering application

Efficient global robust optimization of unconstrained problems affected by parametric uncertainties

“…These problems have been optimized using saddle-point optimization in (Rustem and Howe 2002) while a Kriging-based optimization approach has been used in (Marzat et al 2012). Problem f 1 to f 3 are 4 dimensional with 2 dimensions in X c and 2 dimensions in X e .…”

“…The same problems have also been tested by (Marzat et al 2012(Marzat et al , 2013. The problems have a relatively smaller size, with a maximum of 4 dimensions.…”

“…The reference results for function f 1 to f 7 have been solved to theoretically global robust optimality (Rustem and Howe 2002). Amongst the algorithms in literature, the MiMareK 1 and MiMareK 2 algorithms (Marzat et al 2012;2013) converge to the global robust optimum in the lowest number of expensive function evaluations. In this work, we compare the converge speed of EGRO in terms of number of expensive function evaluations required to reach the global robust optimum against currently available methods.…”

“…Table 1 shows the reference results obtained from (Rustem and Howe 2002;Marzat et al 2012). The table shows the robust optimum locations for the control variable and environment variable and the robust optimum object value.…”

“…In recent work (Marzat et al 2012(Marzat et al , 2013, expected improvement has also been applied to perform unconstrained continuous min-max optimization of black-box functions. The robust solution is found by transforming the min-max problem into a nominal constrained optimization problem with constraint relaxation.…”